Nonlocal Constitutive Relation for Steady Granular Flow

Kamrin, Ken; Koval, Georg

doi:10.1103/physrevlett.108.178301

Cited by 396 publications

(335 citation statements)

References 20 publications

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“…However, the anisotropy is lost in this view, and the fact that θ diminishes as d increases when anisotropy is considered suggests that a mean-field model that includes anisotropy is needed. Such a model would be valuable in building a hydrodynamic description of flow that would apply, for example, to slow flow near walls (57,58), a problem for which current descriptions do not include the role of anisotropy (25,59). …”

Section: Resultsmentioning

confidence: 99%

Scaling description of the yielding transition in soft amorphous solids at zero temperature

Lin

Lerner

Rosso

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

248

370

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Yield stress materials flow if a sufficiently large shear stress is applied. Although such materials are ubiquitous and relevant for industry, there is no accepted microscopic description of how they yield, even in the simplest situations in which temperature is negligible and in which flow inhomogeneities such as shear bands or fractures are absent. Here we propose a scaling description of the yielding transition in amorphous solids made of soft particles at zero temperature. Our description makes a connection between the Herschel-Bulkley exponent characterizing the singularity of the flow curve near the yield stress Σ c , the extension and duration of the avalanches of plasticity observed at threshold, and the density P(x) of soft spots, or shear transformation zones, as a function of the stress increment x beyond which they yield. We argue that the critical exponents of the yielding transition may be expressed in terms of three independent exponents, θ, d f , and z, characterizing, respectively, the density of soft spots, the fractal dimension of the avalanches, and their duration. Our description shares some similarity with the depinning transition that occurs when an elastic manifold is driven through a random potential, but also presents some striking differences. We test our arguments in an elastoplastic model, an automaton model similar to those used in depinning, but with a different interaction kernel, and find satisfying agreement with our predictions in both two and three dimensions. M any solids will flow and behave as fluids if a sufficiently large shear stress is applied. In crystals, plasticity is governed by the motion of dislocations (1, 2). In amorphous solids, there is no order, and conserved defects cannot be defined. However, as noticed by Argon (3), plasticity consists of elementary events localized in space, called shear transformations, in which a few particles rearrange. This observation supports that there are special locations in the sample, called shear transformation zones (STZs) (4), in which the system lies close to an elastic instability. Several theoretical approaches of plasticity, such as STZ theory (4) or soft glassy rheology (5), assume that such zones relax independently or are coupled to each other via an effective temperature. However, at zero temperature and small applied strain rate _ γ, computer experiments (6-11) and very recent experiments (12, 13) indicate that local rearrangements are not independent: plasticity occurs via avalanches in which many shear transformations are involved, forming elongated structures in which plasticity localizes. If conditions are such that flow is homogeneous (as may occur, for example, in foams or emulsions), one finds that the flow curves are singular at small strain rate and follow a Herschel-Bulkley law Σ − Σ c ∼ _ γ 1=β (14, 15). These features are reproduced qualitatively by elasto-plastic models (16)(17)(18)(19)(20) in which space is discretized. In these models, a site that yields plastically affects the stress in its surroundi...

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Section: Resultsmentioning

confidence: 99%

Scaling description of the yielding transition in soft amorphous solids at zero temperature

Lin

Lerner

Rosso

et al. 2014

Proc. Natl. Acad. Sci. U.S.A.

248

370

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“…In this configuration, the pressure is constant and given by P wall . While this may not be obvious from the outset, it has been routinely seen in discrete element method (DEM) calculations [7,13]. Balancing moments gives a decaying shear stress field ofτ = τ wall (R i /r) 2 , where τ wall is the shear stress applied to the inner wall.…”

Section: Annular Shear Flow Without Gravitymentioning

confidence: 99%

A finite element implementation of the nonlocal granular rheology

Henann¹,

Kamrin²

2016

Numerical Meth Engineering

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SUMMARYInhomogeneous flows involving dense particulate media display clear size effects, in which the particle length scale has an important effect on flow fields. Hence, nonlocal constitutive relations must be used in order to predict these flows. Recently, a class of nonlocal fluidity models have been developed for emulsions and subsequently adapted to granular materials. These models have successfully provided a quantitative description of experimental flows in many different flow configurations. In this work, we present a finiteelement-based numerical approach for solving the nonlocal constitutive equations for granular materials, which involve an additional, non-standard nodal degree-of-freedom -the granular fluidity, which is a scalar state parameter describing the susceptibility of a granular element to flow. Our implementation is applied to three canonical inhomogeneous flow configurations: (i) linear shear with gravity, (ii) annular shear flow without gravity, and (iii) annular shear flow with gravity. We verify our implementation, demonstrate convergence, and show that our results are mesh-independent.

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“…The idea that self-activated processes induced by stress fluctuations, or mechanical noise, influence the rheology of slow granular flows has been explored in the literature [50,73], and it seems reasonable to assume that angoricity determines the strength of this mechanical noise.…”

Section: Application To Dynamicsmentioning

confidence: 99%

The Statistical Physics of Athermal Materials

Henkes

Daniels

et al. 2015

Annu. Rev. Condens. Matter Phys.

133

140

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At the core of equilibrium statistical mechanics lies the notion of statistical ensembles: a collection of microstates, each occurring with a given a priori probability that depends only on a few macroscopic parameters such as temperature, pressure, volume, and energy. In this review article, we discuss recent advances in establishing statistical ensembles for athermal materials. The broad class of granular and particulate materials is immune from the effects of thermal fluctuations because the constituents are macroscopic. In addition, interactions between grains are frictional and dissipative, which invalidates the fundamental postulates of equilibrium statistical mechanics. However, granular materials exhibit distributions of microscopic quantities that are reproducible and often depend on only a few macroscopic parameters. We explore the history of statistical ensemble ideas in the context of granular materials, clarify the nature of such ensembles and their foundational principles, highlight advances in testing key ideas, and discuss applications of ensembles to analyze the collective behavior of granular materials.

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Nonlocal Constitutive Relation for Steady Granular Flow

Cited by 396 publications

References 20 publications

Scaling description of the yielding transition in soft amorphous solids at zero temperature

Scaling description of the yielding transition in soft amorphous solids at zero temperature

A finite element implementation of the nonlocal granular rheology

The Statistical Physics of Athermal Materials

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